Hello eyesofjeremy,
I think all of these fonts are installed on OS X. You can use the Font Book app to inspect them. If you do that for Avenir Next Ultra Light Italic, you should get:
Avenir Next Ultra Light Italic
Avenir Next Ultra Light Italic
PostScript name AvenirNext-UltraLightItalic
Full name Avenir Next Ultra Light Italic
Family Avenir Next
Style Ultra Light Italic
Kind OpenType TrueType
Language Afrikaans, Albanian, Asu, Basque, Bemba, Bena, Bosnian, Catalan, Chiga, Congo Swahili, Cornish, Croatian, Czech, Danish, Dutch, Embu, English, Esperanto, Estonian, Faroese, Filipino, Finnish, French, Galician, German, Gusii, Hungarian, Icelandic, Indonesian, Irish, Italian, Kabuverdianu, Kalenjin, Kamba, Kikuyu, Kinyarwanda, Latvian, Lithuanian, Luo, Luyia, Machame, Makhuwa-Meetto, Makonde, Malagasy, Malay, Maltese, Manx, Meru, Morisyen, North Ndebele, Norwegian Bokmål, Norwegian Nynorsk, Nyankole, Oromo, Polish, Portuguese, Romanian, Romansh, Rombo, Rundi, Rwa, Samburu, Sango, Sangu, Sena, Shambala, Shona, Slovak, Slovenian, Soga, Somali, Spanish, Swahili, Swedish, Swiss German, Taita, Teso, Turkish, Vunjo, Zulu
Script Latin
Version 8.0d5e5
Location /System/Library/Fonts/Avenir Next.ttc
Unique name Avenir Next Ultra Light Italic; 8.0d5e5; 2012-08-06
Manufacturer Linotype GmbH
Designer Adrian Frutiger, Akira Kobayashi
Copyright Copyright © 2004 - 2007 Linotype GmbH, www.linotype.com. All rights reserved.
Trademark Avenir is a trademark of Linotype GmbH registered in the U.S. Patent and Trademark Office and may be registered in certain other jurisdictions.
Description Adrian Frutiger designed Avenir in 1988, after years of having an interest in sans serif typefaces. In an interview with Linotype, he said he felt an obligation to design a linear sans in the tradition of Erbar and Futura, but to also make use of the experience and stylistic developments of the twentieth century. The word Avenir means 'future' in French and hints that the typeface owes some of its interpretation to Futura. But unlike Futura, Avenir is not purely geometric; it has vertical strokes that are thicker than the horizontals, an "o" that is not a perfect circle, and shortened ascenders. These nuances aid in legibility and give Avenir a harmonious and sensible appearance for both texts and headlines. In 2004 Adrian Frutiger and the type director of Linotype GmbH Akira Kobayashi reworked the Avenir and created the Avenir Next for the Platinum Collection. It includes new small caps, newly designed true italics, and a complete new range of condensed weights. Avenir Next is a versatile sans serif family, ready for large and complex projects from books to signage to advertising.
Enabled Yes
Duplicate No
Copy protected No
Glyph count 1,189
That should tell you what you missing. The actual "font-family" name is just "Avenir Next". The "Ultra Light Italic" portion is the font-weight and font-style, respectively. So try something like this:
font-family: "Avenir Next", Avenir, Arial, sans-serif;
font-weight: 100;
font-style: italic